This paper considers two-step generalized empirical likelihood (GEL) estimation and tests with martingale differences when there is a computationally simple $\sqrt n$-consistent estimator of nuisance parameters or the nuisance parameters can be eliminated with an estimating function of parameters of interest. As an initial estimate might have asymptotic impact on final estimates, we propose general $C(\alpha )$-type transformed moments to eliminate the impact, and use them in the GEL framework to construct estimation and tests robust to initial estimates. This two-step approach can save computational burden as the numbers of moments and parameters are reduced. A properly constructed two-step GEL (TGEL) estimator of parameters of interest is asymptotically as efficient as the corresponding joint GEL estimator. TGEL removes several higher-order bias terms of a corresponding two-step generalized method of moments. Our moment functions at the true parameters are martingales, thus they cover some spatial and time series models. We investigate tests for parameter restrictions in the TGEL framework, which are locally as powerful as those in the joint GEL framework when the two-step estimator is efficient.

中文翻译：

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有效的两步广义经验似然估计和鞅差检验

本文考虑两步广义经验似然 (GEL) 估计和当存在计算简单时的鞅差检验$\sqrt n$- 干扰参数的一致估计器或干扰参数可以用感兴趣参数的估计函数来消除。由于初始估计可能对最终估计产生渐近影响，我们建议一般$C(\alpha)$-型变换矩以消除影响，并在 GEL 框架中使用它们来构建对初始估计稳健的估计和测试。随着矩量和参数的减少，这种两步法可以节省计算负担。一个适当构造的感兴趣参数的两步 GEL (TEL) 估计器在渐近上与相应的联合 GEL 估计器一样有效。TGEL 去除了相应的两步广义矩量法的几个高阶偏差项。我们在真实参数下的矩函数是鞅，因此它们涵盖了一些空间和时间序列模型。我们研究了 TEL 框架中参数限制的测试，当两步估计器有效时，这些测试在局部与联合 GEL 框架中的参数限制一样强大。